Note on Expansion of Solids

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On heating materials, temperature rises along with their length, breadth, and thickness. The expansion in the length of an object is called linear expansion while the expansion in length and breadth, i.e. in the area is called the superficial expansion. The expansion in three dimensions i.e. in length, breadth, and thickness of an object is called the cubical expansion.

Expansion

Linear expansion: When a substance in the form of wire or rod is heated, its length increases and this is known as linear expansion. Linear expansion of a road

Let us consider a rod of length l1 at θ10C temperature. If we increase the temperature of the rod by heating to θ20C, it will expand. Let l2 e the length of the rod at θ20C. The increase in the length is given by

Hence, the coefficient of linear expansion is defined as the change in length per unit original length per unit change in temperature.

Now,

Final length = \(l_1 + \Delta l\)

$$=l_1 + l_1 \alpha(\theta_2 -\theta_1) $$

$$=l_1 (1+\alpha(\theta_2 -\theta_1)) \dots(iv) $$, which is the equation for final length of rod after change in temperature.

Superficial expansion of a squareSuperficial expansion: When a body is heated, the area of the body increases and it is known as superficial expansion.

Let us consider metal sheet of area A1 at θ10C temperature. If we increase the temperature of the sheet by heating to θ20C, it will expand. Let A2 be the area of the metal sheet at θ20C. The increase in the area is given by

Hence, the coefficient of superficial expansion is defined as the change in area per unit original area per unit change in temperature.

Now,

Final Area = \(A_1 + \Delta A\)

$$=A_1 + A_1\beta(\theta_2 -\theta_1) $$

$$=A_1 (1+ \beta(\theta_2 -\theta_1)) \dots(iv) $$, which is the equation for final area of metal sheet after change in temperature.

Cubical expansion of a cubeCubical expansion: When a body is heated, the volume of the body increases and it is known as cubical expansion.

Let us consider solid metal cube of volume V1 at θ10C temperature. If we increase the temperature of the solid metal cube by heating to θ20C, it will expand. Let V2 be the volume of the solid metal cube at θ20C. The increase in the volume is given by